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Full text of "Men Of Mathematics"

MEN  OF MATHEMATICS

quite right. After all an honestly edited encyclopaedia is in
general a more reliable source of information about the soar-
ing habits of skylarks than a poem, say Shelley's, on the same
topic.

In such an atmosphere of cloying alleged fact, Cantor's theory
of the infinite - one of the most disturbingly original contribu-
tions to mathematics in the past 2,500 years - felt about as
much freedom as a skylark trying to soar up through an atmo-
sphere of cold glue. Even if the theory was totally wrong - and
there are some who believe it cannot be salvaged in any shape
resembling the thing Cantor thought he had launched - it
deserved something better than the brickbats which were
hurled at it chiefly because it was new and unbaptized in the
holy name of orthodox mathematics.

The pathbreaking paper of 1874 undertook to establish a
totally unexpected and highly paradoxical property of the set
of aft algebraic numbers. Although such numbers have been
frequently described in preceding chapters, we shall state once
more what they are, in order to bring out clearly the nature of
the astounding fact which Cantor proved - in saying 'proved'
we deliberately ignore for the present all doubts as to the
soundness of the reasoning used by Cantor.

If r satisfies an algebraic equation of degree n with rational
integer (common whole number) coefficients, and if r satisfies
no such equation of degree less than n9 then r is an algebraic
number of degree n.

This can be generalized. For it is easy to prove that any root
of an equation of the type

cp* + ^a?""1 +    + c^jx + cn = 0,

in which the c's are any given algebraic numbers (as defined
above), is itself an algebraic number. For example, according to
this theorem, all roots of

- (2 + 5VT7) x + ^90"= 0
are algebraic numbers, since the coefficients are. (The first oo-
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